“For many other instances with millions of cities, solutions can be found that are guaranteed to be within 1% of an optimal tour.” [source: Wikipedia, above]

These estimations are sufficiently accurately for logistics and transport companies to plot highly efficient delivery/pickup routes. (Note that similar algorithms are used in the electronics industry to find efficient routing schemes for complex printed-circuit boards and chip layout.)

However, there is no formal proof providing a 100% reliable mathematical method to solve any and every situation. (The 'brute force' computing algorithms merely iteratively test a variety of pseudo-random solutions at very high speed, and then choose the best answer.)